The Schwinger-Dyson equations [1]}, [2]} of quantum field theory can be derived [3]}, [4]} from the invariance of the functional integration under field translations. The structure of the Schwinger-Dyson equations is determined by the action functional, which is involved in the computation of
the vacuum expectation values of the fields.
Let the action \(S[\phi ]\)  be a function of a set of fields denoted by \(\phi (x) \) . The basic Schwinger-Dyson equation takes the form
\(\left\langle \frac{\delta S[\phi ]}{\delta \phi (x)} \, \phi (y_1) \phi (y_2) \cdots \phi (y_n)\right\rangle =i \sum _{j=1}^n \, \delta (x-y_j) \left\langle \phi (y_1) \cdots \phi (y_{j-1}) \phi (y_{j+1}) \cdots \phi (y_n) \right\rangle \; ,\) 
